Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mptfcl Structured version   Unicode version

Theorem mptfcl 26791
Description: Interpret range of a maps-to notation as a constraint on the definition. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Assertion
Ref Expression
mptfcl  |-  ( ( t  e.  A  |->  B ) : A --> C  -> 
( t  e.  A  ->  B  e.  C ) )
Distinct variable groups:    t, A    t, C
Allowed substitution hint:    B( t)

Proof of Theorem mptfcl
StepHypRef Expression
1 eqid 2438 . . 3  |-  ( t  e.  A  |->  B )  =  ( t  e.  A  |->  B )
21fmpt 5893 . 2  |-  ( A. t  e.  A  B  e.  C  <->  ( t  e.  A  |->  B ) : A --> C )
3 rsp 2768 . 2  |-  ( A. t  e.  A  B  e.  C  ->  ( t  e.  A  ->  B  e.  C ) )
42, 3sylbir 206 1  |-  ( ( t  e.  A  |->  B ) : A --> C  -> 
( t  e.  A  ->  B  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726   A.wral 2707    e. cmpt 4269   -->wf 5453
This theorem is referenced by:  mzpsubmpt  26814  eq0rabdioph  26849  eqrabdioph  26850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465
  Copyright terms: Public domain W3C validator